This is a follow up on this earlier question of mine.
We have the following statements:
(HSO) For every infinite set $X$ there exists an injection $f: X \times X \hookrightarrow X$
(HSU) For every infinite set $X$ there exists an injection $f: [X]^2 \hookrightarrow X$
One can prove (HSO) $\rightarrow $ (HSU) as follows:
Assume (HSO). Since $h: X \to X \times X$, $x \mapsto (x,x)$ is an injection by (HSO) and Cantor-Schröder-Bernstein, $X \approx X \times X$ for every infinite set $X$. By a result of Tarski, the latter implies the Axiom of Choice. Once we have AC we can prove HSU as follows: let $f: X \times X \to X$ be an injection. For each unordered pair $\{x,y\}$ choose $g(\{x,y\})$ from the set $\{f((x,y)), f((y,x))\}$.
The following is an exercise I have been unable to solve:
Exercise 40: Find a simpler proof of the implication (HSO) $\rightarrow$ (HSU) in ZF.
Can someone help me solve this exercise? Thanks for your help.